Boundary regularity and stability for spaces with Ricci curvature bounded below

An extension of Gromov compactness theorem ensures that any family of manifolds with convex boundaries, uniform bound on the dimension and uniform lower bound on the Ricci curvature is precompact in the Gromov-Hausdorff topology. In this talk, we will describe how boundaries degenerate in the limit by studying the topological and rectifiable structure of limit spaces. This is based on a joint work with Aaron Naber and Daniele Semola.